After a childhood in karting, Saavedra drove in Formula BMW USA for Gelles Racing, finishing 11th in points. He also drove in two races in Formula BMW ADAC and competed in the Formula BMW World Final, finishing 32nd. In 2007 he switched teams in Formula BMW USA to Eurointernational and captured one win and finished 12th in points despite only competing in 8 of 14 races. He also competed in 6 Formula BMW Asia races capturing 3 wins, two races in Formula BMW ADAC, and finished 4th in the Formula BMW World Final. In 2008 he moved to ATS Formel 3 Cup and finished 2nd in points with 3 wins for HS Technik Motorsport. He also competed in a number of Austria Formula 3 Cup races and finished 9th in points.

He returned to the series in 2010 for Bryan Herta Autosport and made his IndyCar Series debut in the 2010 Indianapolis 500 driving for the same team.[1] Saavedra was on the bubble during most of qualifying during bump day, and crashed during practice laps while waiting for other drivers to attempt to qualify. While he was in the hospital being checked out, and with no working car to attempt to re-qualify, Saavedra appeared out of the race when Tony Kanaan bettered his time knocking him out of the field. However, when both Paul Tracy and Jay Howard withdrew their successful qualifying times and failed to re-qualify (both felt they would be knocked out by other drivers if they did not improve), Saavedra improbably ended up back in position thirty-three and in the race when time ran out for attempts.[2] Saavedra crashed out in a single car accident on lap 159 and was credited with 23rd place in the race. The following month, Saavedra captured Bryan Herta Autosport's second Indy Lights win in the AvoidTheStork.com 100 at Iowa Speedway.

On September 4, 2010, Saavedra officially terminated his contract with Bryan Herta Autosport due to the team's inability to meet contract standards after a string of car failures and negative results.

For 2014, Saavedra signed with KV Racing Technology, racing under the KV/AFS banner. In the inaugural Grand Prix of Indianoplis, Saavedra qualified 1st with a time of 01:23.8822. He stalled when he was starting and he collided with Carlos Muñoz and Mikhail Aleshin on the first lap. His best result throughout the year was 9th in Long Beach.

In 2015, Saavedra joined Chip Ganassi Racing split time in the No. 8 with Sage Karam.[4] Saavedra ran the Indy 500 in a fifth car entered by Ganassi with the No. 17. During the race, Saavedra collided with Jack Hawksworth and hit the Turn 4 wall. After bouncing off, his car was hit just in front of the driver's cockpit by Stefano Coletti. Saavedra was unable to leave the accident under his own power; Saavedra's boot had to be cut off to remove him from the car, after which he was carried by the safety team into an ambulance.